It may be that I'm just not used to JRPG mechanics and tropes at all since this is the first one I've played.
For a first JRPG it might be a little tough.
Off the top of my head.
You want to try and get as many SP restorative items as you can, in a dungeon you can generally farm these by repeating the earlier/easier floors as they're often found in chests, some vending machines sell them too. Try not to use your SP on magic outside of healing unless you really need to. Working on the hermit social link will be helpful to you in the future in this regard also, which is the Fox you meet at the shrine.
As for fusing, your best bet is to just experiment as much as you can. It's a bit of a trial by error unless you follow a guide, but you can't really go wrong too badly. Almost all the enemies you face will have a weakness, while your weakness will be dictated by the persona you have equipped. You want to try and compliment the abilities of your team mates, for example Chie uses ice magic early on and is weak against wind, while Yosuke uses wind and is weak to lightning. So having Persona's on your MC who can switch between element types they don't have will be helpful, remember you can switch them in battle for the MC also.
Your social links will also play into the fusing process, so lets say you spend a lot of time hanging out with Yosuke? You will get bonus points towards your relationship with him if you hang out with him while having a magician persona available to use by your MC. You also gain benefits from your social link with Yosuke when you go to fuse magician persona which are dependent on your social link rank. The social link rank can also give some characters extra abilities in combat.
Your stats can have an effect on what you can do in the world and how much success you'll have doing them, for example the aforementioned Fox at the Shrine needs a certain stat to be a little higher than what it starts at for you to actually meet him.
The dungeons do need to be finished within a certain time period (a set amount of days tied to the weather), you do have a fair amount of time but if you take too long you'll fail the game and be given the option to go back in time by X amount of days to try again. When doing them be sure to buy plenty of Goho-M's while in town so you can go back to the entrance to save regularly when taking on a castle. When you return to it you'll be able to select the floor you left on, but as mentioned earlier sometimes it's worth running around on the lower floors for a bit to grind items/xp/cash. The items you find can be sold to the blacksmith in town, which will open up new weapons and armour for you to buy.
Also, when fighting a boss they'll sometimes give an indication that they're going to do a big attack. If they do you probably want to guard with everyone in order to survive the encounter.
If it's too much for you, when in game bring up the settings menu and it'll give you the ability to tailor the difficulty to something that suits you better. You can change encounter rates, how much XP and cash you get, whether or not you can retry a failed fight etc.
, as I've said previously the story is tip top and totally worth taking in
